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Theorem sotri2 5987
Description: A transitivity relation. (Read 𝐴𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ (𝑆 × 𝑆)
Assertion
Ref Expression
sotri2 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri2
StepHypRef Expression
1 soi.2 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5616 . . . 4 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
32simpld 495 . . 3 (𝐵𝑅𝐶𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 5500 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐵𝑆𝐴𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
64, 5mpan 686 . . . . . 6 ((𝐵𝑆𝐴𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
76con2bid 356 . . . . 5 ((𝐵𝑆𝐴𝑆) → ((𝐵 = 𝐴𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴))
8 breq1 5066 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
98biimpd 230 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
104, 1sotri 5985 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110ex 413 . . . . . 6 (𝐴𝑅𝐵 → (𝐵𝑅𝐶𝐴𝑅𝐶))
129, 11jaoi 853 . . . . 5 ((𝐵 = 𝐴𝐴𝑅𝐵) → (𝐵𝑅𝐶𝐴𝑅𝐶))
137, 12syl6bir 255 . . . 4 ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶)))
1413com3r 87 . . 3 (𝐵𝑅𝐶 → ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
153, 14mpand 691 . 2 (𝐵𝑅𝐶 → (𝐴𝑆 → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
16153imp231 1107 1 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2107  wss 3940   class class class wbr 5063   Or wor 5472   × cxp 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-po 5473  df-so 5474  df-xp 5560
This theorem is referenced by:  supsrlem  10522
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